The generator matrix

 1  0  0  0  1  1  1  2  1  1 X^2+X+2 X^2+X  1 X^2+X+2  1  2  1  1  X  1  0  X  1  1 X^2  1  X X^2  1  1  1  1  1  1
 0  1  0  0  2  1  3  1 X^2+X+2 X+2 X^2+X  1 X+3  1 X+1  1 X^2+3  2  1 X+1 X^2 X^2+2  1 X^2+2 X^2+X  X  1  1 X^2  X X^2+X X+1 X+3 X^2+2
 0  0  1  0  3  1  2  3  0  1  1  1 X^2 X^2 X^2+X+3  X  X X^2+1 X+3  3  1 X^2+X X^2+X X+1  1 X^2+X+2 X+2 X^2+X X^2 X+3 X+1 X+3 X^2+X+2  X
 0  0  0  1  1  2  3  3 X^2+X+1  X  1 X+2 X^2+X X^2+3  3 X^2+1 X^2+X+2 X+1 X^2+X+2 X^2+3 X^2+X  1 X+3  0 X^2+X+1  1 X^2+3  X  X X^2+X+3 X^2+X+2  X X^2+3 X^2+X+1

generates a code of length 34 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 29.

Homogenous weight enumerator: w(x)=1x^0+626x^29+2430x^30+4448x^31+7956x^32+10902x^33+12578x^34+11326x^35+8240x^36+4092x^37+2050x^38+688x^39+125x^40+42x^41+22x^42+2x^43+6x^44+2x^45

The gray image is a code over GF(2) with n=272, k=16 and d=116.
This code was found by Heurico 1.16 in 16.4 seconds.